# Simultaneous Embeddability of Two Partitions

###### Abstract

We study the simultaneous embeddability of a pair of partitions of
the same underlying set into disjoint blocks. Each element of the set is mapped
to a point in the plane and each block of either of the two
partitions is mapped to a region that contains exactly those points that belong to the elements in
the block and that is bounded by a simple closed curve. We establish three main classes of simultaneous
embeddability (*weak*, *strong*, and *full*
embeddability) that differ by increasingly strict well-formedness
conditions on how different block regions are allowed to intersect.
We show that these simultaneous embeddability classes are closely
related to different planarity concepts of hypergraphs. For each
embeddability class we give a full
characterization. We show that (i) every pair of partitions has a weak simultaneous
embedding, (ii) it is \NP-complete to decide the existence of a
strong simultaneous embedding, and (iii) the existence of a full
simultaneous embedding can be tested in linear time.

## 1 Introduction

Pairs of partitions of a given set of objects occur naturally when
evaluating two alternative clusterings in the field of data analysis and data mining.
A *clustering* partitions
a set of objects into *blocks* or *clusters*, such that
objects in the same cluster are more similar (according
to some notion of similarity) than objects in different clusters.
There are a multitude of clustering algorithms that use, e.g., an underlying graph structure or
an attribute-based distance measure to define similarities.
Many
algorithms also provide configurable parameter settings.
Consequently, different algorithms return different
clusterings and judging which clustering is the most meaningful with
respect to a certain interpretation of the data must be done by a
human expert.
For a structural comparison of two clusterings several
numeric measures exist [23], however, a single numeric value
hardly shows where the clusterings agree or disagree. Hence, a data
analyst may want to
compare different clusterings
visually, which motivates the study of simultaneous embeddability of
two partitions.

We provide fundamental characterizations and
complexity results regarding the simultaneous embeddability of a pair of
partitions. While simultaneous embeddability can generally be
defined for any number of partitions, we
focus on the basic case of embedding *two* partitions, which is
also the most relevant one in the data analysis application.
We propose to
embed two alternative partitions of the same set
into the plane by mapping each element of to a unique point and
each block (of either of the two partitions) to a region bounded by a
simple closed curve. Each block region must contain all points that
belong to elements in that block and no point whose element belongs to a
different block.
Hence, in total, each point lies inside two block regions.

A simultaneous embedding of two partitions shares certain properties
with set visualizations like Euler or Venn
diagrams [7, 11, 22]. Its readability
will be affected by well-formedness
conditions for the intersections of the different block regions.
Accordingly, we define a (strict) hierarchy of embeddability classes based on
increasingly
tight well-formedness conditions: *weak*, *strong*, and *full* embeddability.
We show that (i) any two partitions
are weakly embeddable, (ii) the decision problem for strong embeddability is \NP-complete, and (iii) there is a linear-time decision algorithm for full
embeddability. We fully
characterize the embeddability classes in terms of the existence of a
planar support
(strong embeddability) or in terms of the
planarity of the bipartite map
(full
embeddability). Interestingly, both concepts are closely related to
hypergraph embeddings and different notions of hypergraph planarity.
Our \NP-completeness result implies that
vertex-planarity testing of 2-regular
hypergraphs is also \NP-complete.

### 1.1 Related Work

In information visualization there are a large variety of techniques for visualizing clusters of objects, some of which simply map objects to (colored) points so that spatial proximity indicates object similarity [5, 16], others explicitly visualize clusters or general sets as regions in the plane [8, 22]. These approaches are visually similar to Euler diagrams [7, 11], however, they do not give hard guarantees on the final set layout, e.g., in terms of intersection regions or connectedness of regions, nor do they specifically consider the simultaneous embedding of two or more clusterings or partitions.

*Clustered planarity* is a concept in graph drawing that combines a planar graph layout with a drawing of the clusters of a single hierarchical clustering. Clusters are represented as regions bounded by simple closed and pairwise
crossing-free curves. Such a layout is called *c-planar* if no
edge crosses a region boundary more than once [10].

The simultaneous embedding of two planar graphs on the same vertex set is a topic that is well studied in the graph drawing literature, see the recent survey of Bläsius et al. [1]. In a simultaneous graph embedding each vertex is located at a unique position and edges contained in both graphs are represented by the same curve for both graphs. The remaining (non-shared) edges are embedded so that each graph layout by itself is crossing-free, but edges from the first graph may cross edges in the second graph.

Some of our results and concepts in this paper can be seen as a generalization of simultaneous graph embedding to simultaneous hypergraph embedding if we consider blocks as hyperedges: all vertices are mapped to unique points in the plane and two hyperedges, represented as regions bounded by simple closed curves, may only intersect if they belong to different hypergraphs or if they share common vertices. Several concepts for visualizing a single hypergraph are known [14, 18, 15, 4, 3], but to the best of our knowledge the simultaneous layout of two or more hypergraphs has not been studied.

### 1.2 Preliminaries

Let be a finite universe. A
*partition* of groups the
elements of into disjoint *blocks*, i.e., every element is contained in exactly one block . In
this paper, we consider pairs of
partitions of the same universe , i.e., each element is
contained in one block of and in one block of
. In the following
we often omit to mention explicitly.

Let be a collection of subsets of . An
*embedding* of maps every element to a distinct point and every set to a simple, bounded, and closed region such that if and only
if . Moreover, we require that each contiguous intersection
between the boundaries of two regions is in fact a *crossing
point* , i.e., the local cyclic order of the
boundaries alternates around . A *simultaneous embedding*
of a pair of partitions is
an embedding of the union of the
two partitions. We define as the *block region*
of a block and denote its boundary by . Figure 1 shows examples of
simultaneous embeddings in the three different embedding classes to be
defined in Section 2.

A simultaneous embedding induces a subdivision of the plane
and we can derive a plane multigraph by introducing a node
for each intersection of two boundaries and an edge for each section
of a boundary that lies between two intersections. Furthermore, a
boundary without intersections is replaced by a node with a self loop
nested inside its surrounding face. We call the
*contour graph* of and its dual graph the
*dual graph* of . The faces of belong to
zero, one, or two block regions. We call a face that belongs to no
block region a *background face*, a face that belongs to a single
block region a *linking face*, and a face that belongs to two
block regions an *intersection face*. Only intersection faces
contain points corresponding to elements in the universe, and no two
faces of the same type are adjacent in the contour graph.

Alternatively, the union of the two partitions can also be seen as a
hypergraph , where every
element is a vertex and every block defines a
hyperedge, i.e., a non-empty subset of . The hypergraph is *2-regular* since every vertex is
contained in exactly two hyperedges. We denote as the
*corresponding hypergraph* of the pair of partitions
.

*Hypergraph supports* [15] play an important role in
hypergraph embeddings and their planarity. A support of a hypergraph
is a graph on the vertices of
, such that the *induced subgraph* of every hyperedge
is connected. We extend the concept of supports
to pairs of partitions, i.e., we say that a graph is a
support for , if it is a support of
.

We call a support *path based*, if the induced subgraphs of all
hyperedges are paths,^{1}^{1}1Brandes et al. [3] used
a slightly different definition and called a support *path
based* if the induced subgraph of each hyperedge has a Hamiltonian
path. and *tree based*, if all hyperedge-induced
subgraphs are trees, i.e., they do not contain any cycles. For any support
of a pair of partitions we can
always create a tree-based support by removing edges from cycles:
Suppose there exists a block such that
contains a cycle .
If the vertices in are also contained in a common block of
,
we can just remove a random edge from without
destroying the support property.
Otherwise, we can remove an
edge from that connects vertices in two different blocks of
without destroying the support property.

The *bipartite map* of a hypergraph
is defined as the bipartite graph
that has a node for each vertex in and for each hyperedge
in [24]. A node is adjacent to a
node if . We say that is the
bipartite map of a pair of partitions if .

Finally, we define the *block intersection graph* as the
graph with vertex set and edge
set . Thus has
a vertex for each block and an edge between any two blocks that share
a common element. Since only blocks of different partitions can
intersect, we know that is bipartite.

## 2 The Main Classes of Embeddability

We define three main concepts of simultaneous embeddability for pairs of partitions. We will see that these concepts induce a hierarchy of embeddability classes of pairs of partitions.

### 2.1 Weak Embeddability

We begin with *weak embeddability*, which is the
most general concept.

###### Definition 1 (Weak Embeddability)

A simultaneous embedding of two partitions is *weak* if
no two block regions of the same partition intersect. Two
partitions are *weakly embeddable* if they have a weak
simultaneous embedding.

Prohibiting intersections of block regions of the same partition is our first well-formedness condition. A weak embedding emphasizes the fact that the blocks in each partition are disjoint. Since the blocks of any partition are disjoint by definition, it is not surprising that any pair of partitions is weakly embeddable (see Fig. 1(a) for an example).

###### Theorem 2.1

Any two partitions of a common universe are weakly embeddable on any point set.

###### Proof

A spanning forest (in fact, any planar graph) on nodes can always be drawn in a planar way on any fixed set of points in the plane [19]. Let now be a partition. We choose arbitrary, but distinct points in the plane for the elements of . We then generate a spanning tree on the elements in each block and embed the resulting forest in a planar way on the points. Slightly inflating the thickness of the edges of the trees yields simple bounded block regions. We can do this independently for a second partition on the same points and obtain a weak simultaneous embedding.∎

Although the concept of weak embedding does not seem to provide interesting insights into the structure of a given pair of partitions, it guarantees at least the existence of a simultaneous embedding for any pair of partitions that is more meaningful than an arbitrary embedding. An obvious drawback of weak embeddings is that the block regions of disjoint blocks are allowed to intersect, as long as both blocks belong to different partitions—even if they do not share common elements.

### 2.2 Strong Embeddability

Following the general idea of Euler diagrams [7], which do not show regions corresponding to empty intersections, we establish a stricter concept of embeddability. In a strong embedding block regions may only intersect if the corresponding blocks have at least one element in common, and even more, each intersection face of the contour graph must actually contain a point, see Fig. 1(b). This is our second well-formedness condition.

###### Definition 2 (Strong Embeddability)

A simultaneous embedding of two partitions is
*strong* if each intersection face of the corresponding contour
graph contains a point for some . Two
partitions are *strongly embeddable* if they have a strong
simultaneous embedding.

Obviously, a strong embedding is also weak, since blocks of the same partition have no common elements, and thus, cannot form intersection faces. The class of strongly embeddable pairs of partitions is characterized by Theorem 2.2; we show in Section 3 that deciding the strong embeddability of a pair of partitions is \NP-complete.

###### Theorem 2.2

A pair of partitions of a common universe is strongly embeddable if and only if it has a planar support.

###### Proof

Let be a pair of partitions
and let be the contour graph resulting from a strong
embedding of . We construct
a planar support of along
as follows.
First recall that the elements of the universe, which correspond to
the nodes in a support, are represented in by points that
are drawn inside intersection faces. Vice versa,
since is strong, each intersection face contains at least
one point. Hence, we choose one point in each intersection face as
the *center* of this face.
We now create a dummy vertex for each linking face (observe that one block region may induce several linking faces) and link it to the centers of all adjacent intersection faces. The resulting graph is a subgraph of the dual graph of the contour graph and therefore planar. We now connect all remaining vertices in a star-like fashion to the center of their intersection face, routing the edges in a non-crossing way.
We finally remove the dummy vertices by merging them to an adjacent center, linking all adjacent vertices to that center. This graph remains planar. It also has the support property, since all intersection and linking faces of any block region are connected into a single component, and with them all vertices of that block region.

Now we construct a strong embedding from a planarly embedded support of . To this end, we first construct a tree-based support by deleting edges from cycles as described in Section 1.2. Then, we simply inflate the edges of each block-induced subtree. Since the underlying support is embedded in a planar way, this yields a simple block region for every block in such that two block regions only intersect at the positions of the nodes. Hence, the constructed block regions together with the nodes of the support form a strong embedding of . We note that the support graph as a planar graph can in fact be embedded on any point set [19]. Hence, a strongly embeddable pair of partitions can be strongly embedded on any point set.∎

### 2.3 Full Embeddability

In a strong embedding, a single block region may still cross other block regions and intersect the same block regions several times forming distinct intersection faces—as long as each intersection face contains at least one common point.
The last of our three embeddability classes prevents this behavior and requires that the block regions form a collection of *pseudo-disks*, i.e., the boundaries of every pair of regions intersect at most twice and the boundaries of two nested regions do not intersect.
See Fig 1(c) for an example.
This implies in particular that every block intersection is connected, which is a well-formedness condition widely used in the context of Euler diagrams [7], and that block regions do not cross and are thus more locally confined.

###### Definition 3 (Full Embeddability)

A simultaneous embedding of two partitions is
*full* if it is a strong embedding and the regions form a collection of pseudo-disks. Two partitions are fully embeddable if they
have a full simultaneous embedding.

Using a linear-time algorithm for planarity testing [13], the following characterization of fully embeddable pairs of partitions directly implies a linear-time algorithm for deciding full embeddability.

###### Theorem 2.3

A pair of partitions of a common universe is fully embeddable if and only if its bipartite map is planar.

###### Proof

Let be a pair of
partitions and the dual graph of a corresponding full
embedding . The bipartite map of can be constructed as follows. Remove all vertices
(and incident edges) from that stem from a background
face in the contour graph (Fig. 2(a) and
Fig. 2(b)). This results in a planar graph with a set of
*red* nodes resulting from intersection faces, a set of *green* nodes
resulting from linking faces and edges indicating that two faces in
the contour graph are adjacent. Our definitions of simultaneous
embedding and contour graph ensures that nodes of the same color are
not adjacent. Hence, the graph constructed so far is
bipartite. Moreover, each red node is adjacent to at most two green
nodes and, since is full (more precisely, since each
intersection face in the contour graph is adjacent to at most two
linking faces), two green nodes have at most one red common neighbor.
From the same fact it further follows that each block region
in induces at most one linking face in the contour graph,
and thus, the number of green nodes is at most the number of blocks
in .

If there are fewer green nodes than blocks in , at least one block region in must be completely contained in another block region resulting in a red node for the intersection but no green node which could be considered as a representative of the nested block. Hence, the red node representing the intersection is only adjacent to one green node, namely the green node resulting from the linking face of the block region that contains the nested block. Note that no three block regions can be nested, since each point in is contained in exactly two block regions. In this case, we link an additional green node to the red node of the nested block region such that in the end each block in corresponds to a green node and each red node is adjacent to exactly two green nodes. Such an additional leaf obviously preserves planarity (Fig. 2(b)).

Since the bipartite map of a hypergraph (besides the nodes representing blocks) consists of nodes representing the elements in the universe, we finally replace each red node by a set of nodes representing the elements that have been mapped by into the corresponding intersection face, and connect each of these new nodes along the previous edges to the two green nodes previously adjacent to the replaced red node. This again preserves planarity of the finally resulting bipartite map of (Fig. 2(c)).

Now assume a planar embedding of the bipartite map of with green nodes representing the blocks and black nodes representing the elements of the universe. In order to construct a full embedding of , we first construct for each block in a subgraph of the bipartite map such that each subgraph contains the green node that represents the corresponding block and two subgraphs share exactly one black node if and only if the corresponding blocks share at least one element. Since the map of is bipartite, each of these subgraphs is a star with black leaves linked to a green center. Together these stars form a planar subgraph of the bipartite map such that slightly inflating the edges in each star yields simple block regions that intersect exactly at the positions of the black nodes. In the resulting contour graph, two block regions thus intersect at most once and each intersection face is adjacent to exactly two linking faces, each representing a block. A nested block in results in a star that only consists of a green center and one black leaf. In order to completely satisfy the condition of a full embedding, we shrink the block regions of nested blocks such that the boundary of the inner block does not intersect the boundary of the outer block. Deleting the green and black nodes and drawing a set of points that represent the common elements of the intersecting blocks in each intersection face finally yields a full embedding.

We note that our construction uses only a single representative element per block intersection. Thus, in contrast to weak and strong embeddings, it is not clear whether a fully embeddable pair of partitions permits a full embedding on any set of points.∎

### 2.4 Hierarchy of Embeddability Classes

A full embedding is strong by definition and we have seen above that a strong embedding is also weak. Hence, the three embeddability classes introduced in this section induce a hierarchy of embeddability classes. We now show that this hierarchy is strict.

The left side of Fig. 3(a) shows a strong embedding of a pair of partitions that is not fully embeddable. The dotted lines indicate a planar support proving the strong embeddability of . The fact that is not fully embeddable can be seen by considering the bipartite map of , which is a subdivision of , and thus, is not planar (see right side of Fig. 3(a)). The claim then follows from Theorem 2.3.

In order to prove that the class of strong embeddability is a proper subclass of weak embeddability, we take a detour via string graphs.

A graph is a *string graph* if there exists a set of curves in the plane such that if and only if .
Deciding whether a graph is a string graph is \NP-hard [20].
However, Schaefer and S̆tefankovic̆ [21] showed that a graph is
*no* string graph if it is constructed from a non-planar graph by
subdividing each edge at least once.
Together with the following lemma we can thus prove that the pair of partitions
shown in Fig. 3(b) is not strongly embeddable.

###### Lemma 1

The block intersection graph of a strongly embeddable pair of partitions is a string graph.

###### Proof

Let be a strong embedding of a pair of partitions . Our goal is to construct a set of curves in the plane, which correspond to the blocks in such that if and only if and share a common element. This is equivalent to the assertion that the block intersection graph is a string graph. We construct along as follows. First we delete the points in . Then we delete one point of the boundary of each block region that is not an intersection point. This results in a set of curves that correspond to the blocks in , and since is strong, two curves have a common point if and only if the corresponding blocks have at least one common element and the previous block regions were not nested in . For blocks whose block regions are nested in , we replace the curve that represents the nested block by a curve that crosses the surrounding block curve. This finally yields the desired set of curves. ∎

Now consider the pair of partitions in Fig. 3(b). The left side of Fig. 3(b) shows a weak embedding while the right side shows the corresponding block intersection graph, which is constructed from by subdividing each edge exactly once. Consequently, since is not planar, it is no string graph (according to Schaefer and S̆tefankovic̆ [21]). Applying Lemma 1 finally proves that the pair of partitions depicted in Fig. 3(b) is not strongly embeddable.

### 2.5 Embeddability and Hypergraph Planarity

The weak
embeddability class forms the basis of the hierarchy and contains
all pairs of partitions. The strong embeddability class and
the full embeddability class are characterized by the existence of a
planar support and the planarity of the bipartite map of a pair of
partitions, respectively, where the latter directly implies a linear
time algorithm for the corresponding decision problem. Moreover,
these characterizations reveal close relations to the hypergraph planarity
concepts of *Zykov* and *vertex planarity*.

A hypergraph is Zykov-planar [25], if there exists a subdivision of the plane into faces, such that each hyperedge can be mapped to a face of the subdivision, and each vertex can be mapped to a point on the boundary of all faces that represent a hyperedge containing . Walsh [24] showed that a hypergraph is Zykov planar if and only if its bipartite map is planar.

In contrast, a hypergraph is
vertex-planar [14] if there exists a subdivision of the
plane into faces, such that every vertex can be mapped to a
face and for every hyperedge , the interior of the
union of all faces of the vertices in is connected. Kaufmann et
al. [15] showed that a hypergraph is vertex planar if and
only if it has a planar support.
This shows that the class of *fully*
embeddable pairs of partitions is a subclass of Zykov planar
hypergraphs, and the class of *strongly*
embeddable pairs of partitions is a subclass of vertex planar
hypergraphs.

## 3 Complexity of Deciding Strong Embeddability

In this section we show the \NP-completeness of testing strong embeddability. As a consequence, testing whether the corresponding hypergraph of a pair of partitions has a planar support is also \NP-complete by Theorem 2.2. This seems not very surprising considering the more general hardness results of Johnson and Pollak [14] and Buchin et al. [4] who showed that deciding the existence of a planar support and a 2-outerplanar support in general hypergraphs is \NP-hard. However, we consider a restricted subclass of 2-regular hypergraphs, thus, the \NP-hardness of our problem does not directly follow from the previous results. Moreover, other special cases, e.g., finding path, cycle, tree, and cactus supports are known to be solvable in polynomial time [14, 4, 2]. Together with the characterization of Theorem 2.2, Theorem 3.1 immediately implies that testing the vertex planarity of a 2-regular hypergraph is \NP-complete.

###### Theorem 3.1

Deciding the strong embeddability of a pair of partitions is \NP-complete.

The existing hardness results [14, 4] rely on
elements that are contained in more than two hyperedges and could not be
adapted to our 2-regular setting. Instead we prove the hardness of deciding
strong embeddability by a quite different reduction from the
\NP-complete problem monotone planar 3Sat [9].
A monotone planar 3Sat formula is a 3Sat formula whose
clauses either contain only positive or only negated literals (we call
these clauses *positive* and *negative*) and whose
variable-clause graph is planar. A *monotone
rectilinear representation* (MRR) of is a drawing of
such that the variables correspond to axis-aligned rectangles on the
x-axis and clauses correspond to non-crossing E-shaped “combs” above
the x-axis if they contain only positive variables and below the
x-axis otherwise; see
Fig. 6(a).

An instance of monotone planar 3Sat is an MRR of a monotone planar 3Sat formula . In the proof of Theorem 3.1 we will construct a pair of partitions that admits a strong embedding if and only if is satisfiable.

For the sake of simplicity, we restrict
the class of
strong embeddings to the subclass of *proper strong embeddings*, which
is equivalent, as we can argue that a pair of
partitions has a strong embedding if and only if it also has a proper one.
A strong embedding is *proper* if the contour
graph does not contain background or linking faces that are
adjacent to only two other faces.
Figure 4 illustrates how background or linking faces violating this condition can be removed, transforming a strong embedding into a proper one.

We say that two proper strong embeddings are *equivalent* if the
embeddings of their contour graphs are equivalent, i.e. if the cyclic order of the edges around each vertex is the same. A pair of
partitions has a *unique strong embedding* if all proper strong
embeddings are equivalent.
Note that, analogously to the definition of equivalence of planar graph embeddings, two equivalent proper strong embeddings may have different unbounded outer background faces.
Our construction in the hardness proof is independent of the choice of the outer face.

Next we define a special pair of partitions that has a unique grid-shaped embedding as a scaffold for the gadgets in the subsequent proof of Theorem 3.1. The first step is to construct a base graph for two integers and . The graph is a grid with columns and rows of vertices with integer coordinates for and . Each vertex with coordinates is connected to the four vertices at coordinates (if they exist). Between the middle rows and we remove all vertical edges except for those in columns . This defines larger grid cells of width in this particular row. Figure 5 (left) shows an example.

From we construct a pair of partitions as follows (see Fig. 5). For each
vertex with coordinates we create a *vertex
block* in partition . For each edge in we create a
chain of four *edge blocks* , , ,
, such that and are in the same
partition as and and are in the same
partition as .
We distribute five distinct elements among the edge blocks of
and the vertex blocks for and such that they form the desired chain pattern and each intersection face contains one common element. The
pair is indeed a pair of partitions
as every element belongs to
exactly one block of each partition. Edge blocks contain two and
vertex blocks up to four elements (depending on the degree of the
corresponding vertex in ). Below we will add the gadgets of the reduction on top of , for which it is required that there is an edge block in each partition that does not share any element with a vertex block. This explains why we link blocks of adjacent vertices by chains of four blocks.

The next lemma shows that has a unique embedding, which is a consequence of the fact that is a subdivision of a planar 3-connected graph (assuming ) and thus it has a unique embedding. This property is inherited by in our construction.

###### Lemma 2

The pair of partitions has a unique embedding.

###### Proof

First, we observe that the base graph is a subdivision of a planar 3-connected graph (assuming ) and thus it has a unique embedding (up to the choice of the outer face) in the plane. We claim that this property is inherited by in our construction.

Each edge block contains exactly two elements and intersects exactly two blocks of the other partition. Thus its contour subgraph in any proper strong embedding is isomorphic to the 4-cycle with two non-incident duplicate edges inside, which belong to the boundaries of the two intersecting blocks. Each vertex block contains two, three, or four elements, depending on the degree of in . Since intersects with edge blocks of the other partition there are exactly two intersection points with the boundary of each of these edge blocks in a proper strong embedding (if there were four intersection points, then the edge block would not be proper). Thus the contour subgraph of is a 4-, 6-, or 8-cycle with two, three, or four non-incident duplicate edges inside belonging to the intersecting edge blocks. There is a bijection between the possible cyclic intersection orders of the edge blocks and the possible cyclic orders of the incident edges of vertex in . Thus we have locally the same embedding choices of the contour graph of as for the vertex in . Since has a unique embedding, and since each edge of is represented in by a sequence of four edge blocks with a locally unique embedding between the two incident vertex blocks, we conclude that for every proper strong embedding of the induced contour graph is the same graph with the same unique planar embedding. Otherwise we could derive two different embeddings of , which is a contradiction. ∎

Now we have all the tools that we need to prove our main theorem in this section.

###### Proof (of Theorem 3.1)

First we show that the problem is in \NP. By Theorem 2.2 we know that a pair of partitions is strongly embeddable if and only if it has a planar support. Thus we can “guess” a graph on and then test its planarity and support property in polynomial time. This shows membership in \NP. It remains to describe the hardness reduction.

Let be a planar monotone 3Sat formula together with an
MRR. First we construct the pair of
partitions for the base graph
, where
is the number of clauses of and is the number of variables
of .
By Lemma 2
has a unique proper grid-like
embedding. We call the *base
grid* and the special cells between rows and the
*variable cells* of the base grid.

Next we augment the pair of partitions by additional blocks, one for
each clause, where positive clauses are added to
and negative clauses to .
The definition of these *clause blocks*
closely follows the layout of the
given MRR, see
Fig. 6(a). Let be the positive clauses of
ordered so that if is nested inside the E-shape
of in the given MRR then . Analogously let
be the ordered negative
clauses. We describe the definition of the block
for a positive clause (); blocks
for negative clauses are defined symmetrically. We
create an *intermediate embedding* of
(which is not yet strong but serves as a template for
a later strong embedding) by putting on top of
the base grid^{2}^{2}2The idea of fixing paths to an
underlying grid is inspired by Chaplick et
al. [6]. and adding new
elements to and to certain edge blocks in
. This fixes to run through two
mirrored E-shaped sets of grid cells of our choice
(Fig. 6(b)). In the upper half of the base grid, is assigned
to run between rows and . Furthermore,
is assigned to three columns leading towards the
variable cells from the top. Let be a variable
contained in and assume that is the -th
positive clause from the right connecting to in
the embedding of the given MRR. Then runs
between columns and . In the
lower half of the base grid we translate and mirror
the resulting E-shape as follows. We let occupy
the cells between rows and and
the three columns are shifted to the left by the
number of occurrences of the respective variable in
negative clauses
(Fig. 6(b)). Since each
variable cell is columns wide, we can always
assign each clause to a unique column of in the
top and bottom half of the grid in this way.

We actually
fix to the base grid by adding one shared element for each crossed edge of a grid cell to both
and the respective edge block of that does not share an element with a vertex
block in (recall that contains such a
block in each partition and for each grid edge). No two blocks of the same clause type (positive or negative)
intersect, but blocks of different type do intersect in certain grid cells. For each grid cell shared
between a positive and negative block (except for the variable cells) we add one shared element
(black dots in Fig. 6(b)) and call the respective grid cell the
*home cell* for this element. Recall that the orders of the incoming blocks from the top
and the bottom of each variable cell are inverted. Thus, within each variable cell the blocks of each
pair of a positive and negative clause using the corresponding variable intersect, but no shared
element is added. We denote the resulting new pair of partitions as and observe that its size is polynomial in the size of .

Next we argue about the strong embedding options in contrast to the immediate embedding for a clause block in . In the intermediate embedding each block has three connections through variable cells linking the upper E-shape with the lower E-shape. Any element shared with an edge block of the uniquely embedded base grid must obviously be reached by the block region of . Since the block region must be simple, any strong embedding of results from opening the intermediate embedding of in exactly two grid cells so that the resulting block region of is connected and has no holes. Additionally, a shared element must be placed in any intersection of the block region of with block regions of other clause blocks.

First we assume that is a satisfiable formula and a satisfying variable assignment is given.
We need to show that has a strong embedding. If a variable
has the value *true* in the given assignment we open all blocks of negative clauses using
in the corresponding variable cell; if is *false* we open all blocks of positive
clauses using . Thus no blocks intersect in variable cells any more. If a clause contains more
than one *true* literal, we open all but one connection in its variable cells of *true*
literals. Since the assignment satisfies , we know that each clause block is opened exactly
twice in its variable cells and thus forms a valid simple block region. Moreover, we place all shared
elements in their home cells so that every block intersection contains an element and the
embedding is strong. We call a strong embedding of with the
above properties a *canonical embedding*.

Now assume that has a strong embedding. We know that the base
grid has its unique embedding and that each block is embedded as a simple region that results from
opening the intermediate embedding (with its two E-shapes linked through three variable cells) in
exactly two cells. If the embedding is already canonical, we can immediately construct a satisfying
variable assignment for : if a variable cell is crossed by clause blocks in we
set the variable to *true*, otherwise we set it to *false*. Since every clause block is
connected we know that this assignment satisfies all clauses. If the embedding is not canonical we show
that it can be transformed into a canonical embedding as follows. In a non-canonical embedding it is
possible that two blocks and intersect in a variable cell and have a shared element
in their intersection face in the cell of rather than in the home cell of that element. This
means, on the other hand, that in some shared home cell of and , say in the
upper half, at least one of the two blocks is opened (as there is no more shared element to put into an
intersection face). Thus the grid cell splits the E-shaped block region of one or both blocks
in the upper half into two disconnected components, meaning that each opened block crosses at least two
variable cells in order to connect both components via the lower half. Hence we can safely split any
block that is opened in in the cell of variable , re-connect it inside , and
place the shared element of and into its home cell . This removes the block
intersection in the cell of . Once all block crossings within variable cells are removed, the
resulting embedding is a canonical embedding and we can derive the corresponding satisfying variable
assignment. ∎

## 4 Extensions and Conclusion

We have characterized three main embeddability classes for pairs of partitions, which in fact form a strict hierarchy, and we have shown \NP-completeness of deciding strong embeddability. From a practical point of view the class of strong embeddings is of particular interest: it guarantees that every intersection between block regions is meaningful as it contains at least one element, but on the other hand allows blocks to cross, imposing less restrictions than full embeddings. Interesting subclasses of strong embeddings that further structure the space between strong and full embeddability can be defined and we mention two of them.

In a strong embedding two blocks can intersect many times forming
disjoint intersection faces, whereas a full embedding permits only a
single connected intersection region for any pair of blocks (this is
also a common requirement for Euler diagrams [7]). In
*single-intersection strong embeddings* we adapt this
*unique intersection region* condition of full embeddings, but
still permit that two blocks cross in the embedding. This new
class is a true subclass of strong embeddings, see the example in
Fig. 7. It is
open whether the corresponding decision problem is still
\NP-complete since our proof is based on the existence of multiple
intersection regions between pairs of blocks.

Another interesting subclass are *strong grid embeddings*, in
which the blocks of and are embedded as horizontal and vertical ribbons, respectively,
which intersect in a matrix-like fashion, see
Fig. 7. Obviously a strong
grid embedding is also a single intersection strong embedding, but
again strong grid embeddings form a true subclass, see
Fig. 7. It is easy to see that a pair of
partitions admits a strong grid embedding if and only if its block
intersection graph is a *grid intersection
graph* [12], i.e., an intersection graph of horizontal
and vertical segments in the plane.
Kratochvíl [17] showed that
deciding whether a bipartite graph is a grid-intersection graph is
\NP-complete.
All fully
embeddable pairs of partitions have a strong grid embedding: The
bipartite map of a fully embeddable pair of partitions is planar by
Theorem 2.3 and immediately induces a planar bipartite
block intersection graph, which, according to Hartman et
al. [12], is a grid intersection graph. This implies \NP-completeness of deciding strong grid embeddability. But as the
example of Fig. 3(a) shows, not every instance with
a strong grid embedding admits a full embedding.

It is an interesting direction for future work to study the generalization of our embeddability concepts to partitions. While weak embeddability and its properties extend readily to any number of partitions, it is less obvious how to generalize strong and full embeddability. One possibility is to require the properties in a pairwise sense; otherwise constraints for new types of faces in the contour graph belonging to more than one but less than block regions might be necessary.

On the practical side, future work could be the design of algorithms that find visually appealing simultaneous embeddings according to our different embeddability classes. Finally, if the partitions are clusterings on a graph, one would ideally want to simultaneously draw both the partitions and the underlying graphs.

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